The discovery of gravitational waves emitted by a binary coalescing system has opened new possibilities to probe our Universe and to test General Relativity.
The increase measurements of gravitational waves signals, and the higher experimental precision expected from the next generation of gravitational waves detectors, require a more accurate knowledge of the dynamics of the two-body problem in General Relativity, which has been studied within this thesis following two distinctive, but complementary, perturbative schemes, which are the PN (post-Newtonian) and the PM (post-Minkowskian) expansions. Calculations within the two approaches can be done in an innovative way, by taking advantage of Effective Field Theories and of Scattering Amplitudes multi-loop techniques, that for many years have been developed in Particle Physics, and that only very recently have been applied to this new sector.
The conservative dynamics of a slow inspiral binary can be described by taking deviations from the Newton's two-body potential in terms of Post-Newtonian contributions, where at \(n\) -PN order one considers corrections that scale as \(G_N^{n-l}v^{2l}\)  where \(0\le \ell \le n-1\) . PN physics can be studied using an Effective Field theory approach, where the binary dynamics can be described as a system of point particles which are gravitationally interacting in a flat spacetime. In this approach, PN corrections can be organized in terms of Feynman diagrams, which have a clear power counting in terms of \(v/c\) . Such diagrams, can be topologically mapped into multi-loop two-points and one-point functions, and we took advantage of this property to perform high precision calculations using modern multi-loop techniques. Following the method of regions, the computations can be divided into near and far zone. Focusing on the far-zone dynamics, the binary system can be described as a single source, expanded in terms of multipole moments, and emitting radiation gravitons. Within this framework we considered the so-called Hereditary Effects, which are the effects of the past evolution of a material system on its present gravitational dynamics. They can be described as diagrams containing radiation gravitons emitted by the system and then scattered back into the same GWs source, and they can be divided into back-scattering, tail and memory terms. Moreover, in dealing with radiation problems like this, one should use the so-called Schwinger-Keldysh formalism (or In-In or CTP formalism), instead of Feynman formalism. The first original contribution of this thesis has been the development of an automatic computational algorithm in Mathematica for the evaluation of Hereditary Effects in PN Physics, simultaneously in Feynman formalism and CTP Formalism. Starting from the effective Lagrangian for the theory, the code evaluates the Feynman rules, generates the diagrams and evaluates them using powerful identities, such as Integration-by-parts identities, to decompose difficult Feynman amplitudes appearing in terms of a basis of few integrals, known as Master Integrals, leaving us to the computation of the MIs only. Moreover, we implemented a new way to deal with In-In diagrams, which uses Feynman amplitudes as building blocks for In-In ones, where only the Master Integrals have to be evaluated case-by-case according to the boundary conditions appearing in the propagators of the diagrams. We used this algorithm to evaluate the Hereditary Effects up to 5PN order in both formalism, which involved the evaluation of several 2-loop integrals, and we were able to check our results with those appearing in literature. This powerful algorithm can be easily generalized to compute higher order Hereditary Effects. In literature appear different derivations of the 5PN dynamics, obtained with different techniques, which provide incompatible predictions for physical observables. These differences have to be sorted out, and future works starting from this project can try to solve this puzzle by better understanding the origin of each term appearing, and by comparing them with the ones entering in other computations, paving the way to the derivation of higher order PN effects.
During the second part of this project, we studied a framework to deal with relativistic gravitational scattering processes, which is based on the Post-Minkowskian approach to General Relativity, which is a weak-field expansion only, where at \(n\) -PM order we consider contributions that scale as \(G_N^n\) . By treating General Relativity as the low-energy EFT of a quantum theory of gravity, this formalism gave us the possibility to obtain both classical and quantum contributions to physical observables using Feynman diagrams. With a generalization of the Lippman-Schwinger equation, one can study the interaction potential between two bodies from the Fourier transform of the low-energy limit of PM amplitudes. Within this formalism we studied the classical and quantum corrections to the bending angle of a light-like scalar under the influence of a massive object at 1-loop order, which in terms of scattering amplitudes could be described as a gravitational scattering between a massive and a massless scalar object. The second original contribution of this thesis has been the development of a novel computational algorithm which automatically generates the diagrams, and then computes them in dimensional regularization using multi-loop techniques, and we used it to evaluate the amplitude for this process at 1-loop. Since the bending angle evaluation involves a Fourier transform, it was not necessary to compute the full 1-loop amplitude, but we could focus only on the massless double-cut, since it captures the non-analytic pieces, which are the only ones surviving after the Fourier transform. From the low-energy limit of this amplitude, we obtained the classical and quantum corrections to the bending angle, which was computed using two different techniques, and we found new quantum terms arising which were not covered in previous analyses. As discussed on ChapterREF , the presence of such terms opens the need to review the analyses done so far in [1]}, [2]}, [3]}. In particular, it is necessary to better understand the role of the new logarithmic term arising from the computation, how it is affecting the physical observables, and if the factorization theorems proved in [4]}, [5]} are valid also including such new terms. Considering also the massless particle double-cut [2]} the situation gets even worse, since there is a divergence appearing in the non-analytic part of the amplitude, which makes any physical prediction meaningless. Moreover, the appearance of such contributions can be used to put constrains on Wilson coefficients, considering higher order terms in the EFT of gravity. This analysis can be extended also considering different types of external fields, as already done in [1]}, [5]}, and using modern scattering amplitudes techniques, such as Generalized Unitarity. More in general, one needs to understand which is the physical meaning of the quantum corrections appearing, and such analysis can start only from a full control of the 1-loop quantum amplitude, which can start from this work. In the last part of this master thesis we made use of this 1-loop computation to get another relevant physical result, which is the computation of the 1-loop classical and quantum corrections to the Newtonian potential between two massive scalar fields. Just by making the other field massive, we were able to use the same computational algorithm to get the 1-loop amplitude and the corrections to the potential, finding agreement with previous results appearing in literature. This was an important calculation to check the validity of our computation also for the massless case. Moreover, the computational algorithm can be generalized to higher order in \(G_N\) , and can be used as an additional test for higher order PM calculations.
Recently, a new interesting method has been proposed [9]}, which allows obtaining predictions for classical gravitational waves observables directly from scattering amplitudes, without going through intermediate un-physical steps, such as the evaluation of effective actions and potentials. By analyzing in detail the appearance of classical contributions from quantum mechanical calculations, this new way of thinking about gravity could be leading the sector in the next years.
